Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20R.8 nth Roots and Rational Expressions In this section, we will… Evaluate nth Roots Simplify Radical Expressions Add, Subtract, Multiply and Divide Radical Expressions Rationalize Denominators Simplify Expressions with Rational Exponents Factor Expressions with Radicals or Rational ExponentsChapter R Section 8: nth Roots and Rational ExponentsR.8 nth Roots and Rational Expressions: nth Roots If a is a non-negative real number, any number b, such that is the square root of a and is denotedIf a is a non-negative real number, any non-negative number b, such that is the primary square root of a and is denotedExamples: Evaluate the following by taking the square root.2b a=b a=12116-2b a=b a=The principal root of a positive number is positiveNegative numbers do not have real # square rootsprincipal root:Recall from Review Section 2…R.8 nth Roots and Rational Expressions: nth Roots The principal nth root of a real number a, n > 2 an integer, symbolized by is defined as follows:where and if n is evenwhere a, b are any real number if n is oddExamples: Simplify each expression.3270a �0b � means nna b a b= =nanaindexradicandradical38-481416-51-principal root: if 3 is oddnna a n= � if 2 is evennna a n= �Properties of Radicals: Let and denote positive integers and let a and b represent real numbers. Assuming that all radicals are defined:Simplifying Radicals: A radical is in simplest form when: No radicals appear in the denominator of a fractionThe radicand cannot have any factors that are perfect roots (given the index)Examples: Simplify each expression.2n �2m �n n nab a b=nnna abb=( )mnmna a=1250316R.8 nth Roots and Rational Expressions: Simplify Radical ExpressionsSimplifying Radical Expressions Containing Variables: Examples: Simplify each expression. Assume that all variables are positive.55x8416x7b6 5354x y-When we divide the exponent by the index, the remainder remains under the radicalR.8 nth Roots and Rational Expressions: Simplify Radical ExpressionsR.8 nth Roots and Rational Expressions: Add, Subtract, Multiply and Divide RadicalsAdding and Subtracting Radical Expressions: simplify each radical expression combine all like-radicals (combine the coefficients and keep the common radical)Examples: Simplify each expression. Assume that all variables are positive.125 20-2 12 3 27-2 2 3 338 25 8xy x y x y- +54432 2x x+R.8 nth Roots and Rational Expressions: Add, Subtract, Multiply and Divide RadicalsMultiplying and Dividing Radical Expressions: Examples: Simplify each expression. Assume that all variables are positive.35 20x x234 23381xyx y( )433 10we will use: n n nab a b=we will use: nnaanbb=( )we will use: mnmna a=R.8 nth Roots and Rational Expressions: Add, Subtract, Multiply and Divide RadicalsExamples: Simplify each expression. Assume that all variables are positive.( ) ( )5 8 3 3-( ) ( )2 2x x+ -( )22 3 5+( )4 2 3 5 2 8-Vegas RuleR.8 nth Roots and Rational Expressions: Rationalize DenominatorsRationalizing Denominators: Recall that simplifying a radical expression means that no radicals appear in the denominator of a fraction.Examples: Simplify each expression. Assume that all variables are positive.24554 2342R.8 nth Roots and Rational Expressions: Rationalize DenominatorsRationalizing Binomial Denominators: example:Examples: Simplify each expression. Assume that all variables are positive.23 1+The conjugate of the binomial a + b is a – b and the conjugate of a – b is a + b.23 1+2 13 2 2-+R.8 nth Roots and Rational Expressions: Simplify Expressions with Rational ExponentsEvaluating Rational Exponents: Examples: Simplify each expression.If is a real number and 2 is an integer and assuming that all radicals are defined: a n �1416180If is a real number and and 2 are an integer and assuming that all radicals are defined: a m n �1nna a=( )mnmnmna a a= =3243225-( )23278R.8 nth Roots and Rational Expressions: Simplify Expressions with Rational ExponentsSimplifying Expressions Containing Rational Exponents: Recall the following from Review Section 2:Laws of Exponents: For any integers m, n (assuming no divisions by 0)m n m nx x x+=( )nm mnx x=( )nn nxy x y=nnnx xy y� �=� �� �01x =mm nnxxx-=1nna a=( )mnmnmna a a= =new!new!1nnxx-=nnx yy x-� �� �=� �� �� �� �1nnxx-=andR.8 nth Roots and Rational Expressions: Simplify Expressions with Rational ExponentsExamples: Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive.21 132 4x x x-( )344 8x y( )( )( )1124342 22xy x yx yR.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational ExponentsFactoring Expressions with Radicals and/or Rational Exponents: Recall that, when factoring, we take out the GCF with the smallest exponent in the terms.Examples: Factor each expression. Express your answer so that only positive exponents occur.312 2x x-( ) ( )312 23 3x x+ - +( ) ( )210 1 5 1x x x x+ - +R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational ExponentsExamples: Factor each expression. Express your answer so that only positive exponents occur.( ) ( )4 13 32 244 4 23x x x x+ + � + �R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational ExponentsExamples: Factor each expression. Express your answer so that only positive exponents occur.( )312 26 2 3 8 0x x x x׳ + +R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational ExponentsExamples: Factor each expression. Express your answer so that only positive exponents occur.( ) ( )4 13 322 3 4 4 3 4x x x x+ + � +R.8 nth Roots and Rational Expressions: ApplicationsExample: The final velocity, v, of an object in feet per second (ft/sec) after it slides down a frictionless inclined plane of height h feet is: where is the initial velocityin ft/sec of the object.What is the final velocity, v, of an object that slides down a frictionless inclined plane of height 2 feet with an initial velocity of 4 ft/sec?2064v h v= +0vIndependent Practice You learn math by doing math. The best way to learn math is to practice,